Question: Find the curve defined by the equation
\[r = 4 \tan \theta \sec \theta.\](A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola

Enter the letter of the correct option.
From $r = 4 \tan \theta \sec \theta,$
\[r = 4 \cdot \frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\cos \theta}.\]Then $r \cos^2 \theta = 4 \sin \theta,$ so
\[r^2 \cos^2 \theta = 4r \sin \theta.\]Hence, $x^2 = 4y.$  This is the equation of a parabola, so the answer is $\boxed{\text{(C)}}.$

[asy]
unitsize(0.15 cm);

pair moo (real t) {
  real r = 4*tan(t)/cos(t);
  return (r*cos(t), r*sin(t));
}

path foo = moo(0);
real t;

for (t = 0; t <= 1.2; t = t + 0.1) {
  foo = foo--moo(t);
}

draw(foo,red);
draw(reflect((0,0),(0,1))*(foo),red);

draw((-12,0)--(12,0));
draw((0,-5)--(0,30));
label("$r = 4 \tan \theta \sec \theta$", (22,15), red);
[/asy]